Fat-Tailed Distibutions

December 2024

Fat-tailed distributions are as misunderstood as they are common. It is the notion that only a small percentage of inputs account for a disproportionately large amount of output. For ease of reference, I lump together pareto, log normal, and power law concepts all under this loose banner of “FTDs” despite their statistical differences. 

Here are some of my favourite examples of FTDs: 

Human capital: Jewish intellectuals from Budapest born approximately 20 years either side of 1900 produced a significant amount of humanity’s technological innovations in the mid-late 20th century (dozens of figures such as Von Neumann – computers, Grove –semiconductors, Von Karman – rockets, Teller – physics, etc.). 

Mining: A single mine in Spruce Pine, North Carolina supplies nearly all high purity quartz used to manufacture silicon for advanced integrated circuits. 

Technology: ASML is the only company on earth that can do advanced EUV lithography for <7nm semiconductors. Their most advanced machines cost up to $400m each, and have to be loaded across 20 trucks or 3 Boeing 747s. 

Physics: The Phoenix A supermassive blackhole has a mass of 100 billion solar masses, when most blackholes have a mass of around 5-100 solar masses. 

Finance: The 10 top-performing days for the S&P500 drove more than 40% of the index's total returns from 1980 to 2020 (that's just 0.1% of that period). 

Linguistics: The 100 most used words in English account for 50% of all words used texts. This ratio is achieved with just 50 words in Spanish, and in Chinese Mandarin the most 100 common characters account for 75% of all written text. 

It is also well known that such a pattern occurs in society: that the top 1% have half of the wealth, or that 5% of offenders commit over half of all violent crimes. These observations invite controversy in their proposed diagnoses and solutions.  

So, what drives FTDs? 

Positive feedback loops: the idea that success begets success. The wealthiest will accumulate wealth faster thanks to financial compounding, and – more intangibly – growing their networks, reputation, and opportunities. Larger blackholes will have the ability to gravitationally attract more matter into them, in turn giving them more gravitational attraction with which to further grow. 

System complexity: in complex systems, non-linear interactions drive extreme outcomes. The fact that all of the 8000m+ mountains in the world are in Asia is not because high mountains themselves have the ability to induce height in other mountains immediately around them. It is because a series of specific geological phenomena (e.g. a convergent boundary of two giant plates) have interacted in an aligned manner to produce extreme mountains within the same concentrated part of the world. The same process describes concentrations of certain minerals or resources in small parts of the world. 

Scale invariance: for a system to exhibit a FTD, its components must have the potential to grow or behave similarly across different scales. Patterns must look similar at all sizes. In the case of a stock market, a $1tn public company could very well gain 5% in a single day (just as a much smaller company could do) and add $50bn to their market cap. It is then easy to see how just 10 companies constitute nearly one-third of the S&P500’s market cap. City populations follow the same patterns of non-linear distribution around the world, regardless of size. 

Energy minimisation: Natural phenomena that demonstrate FTDs will be driven by a system tending towards maximising entropy and minimising energy. River systems and clouds both form as the result of non-linear interactions between fluids within physical constraints. The result is the agglomeration of dominant, larger formations which tend towards equilibrium. 

The logical basis for why FTDs occur is undeniable. An entirely separate question begged by this is why an occurrence so fundamental remains so controversial.